Elaboration d'une modélisation mathématique du transfert multi-échelle des signaux mécaniques dans l'os cortical humain. Aspects théoriques et simulations numériques

Understand bone remodelling needs the knowledge of mechanical information transfer: what information receives a cortical bony cell when the bone is solicited? The purpose of this study is the elaboration of a modeling and of mathematical tools allowing to estimate, from a mechanical loading applied on a human bone, various fields existing in collagen, hydroxyapatite and bony fluid.Mathematical theories of homogenization and flow in porous media are used. Model is made by taking successively into account spatial organization of hydroxyapatite crystals, different mineralizations, a new behaviour's law, motion of fluid containing ions in each of architectural levels and homogenization of complex composite structures (lamellae, osteons, cortical bone).On a mathematically point of view, asymptotic developments method in a new piezoelectric framework (with threshold) is used. One establishes all necessary relationships, a property of convergence and a local analysis is made. The return to microscopic level is made directly via a technique of localization or indirectly when the effect of threshold occurs. Developped computational methods have been packed in two softwares.On a biomechanical point of view, it has been established that human cortical bone is a non piezoelectric orthotropic medium for which anisotropy is essentially involved by the nanoscopic architecture, that they are two types of flow in osteon and that flows in the osteons differ according to their architecture. A process being able to explain how cells know what architecture to give to the collagen tissue is thus pointed out.Comprendre le remodelage osseux nécessite la maîtrise du transfert des informations mécaniques: quelles informations reçoit une cellule osseuse de la partie corticale lorsque l'os est sollicité ? Ce mémoire est l'élaboration d'une modélisation et d'outils mathématiques permettant d'estimer, à partir d'un chargement mécanique appliqué à un os humain, divers champs existant dans le collagène, l'hydroxyapatite et le fluide environnant.On utilise les théories mathématiques de l'homogénéisation et des écoulements en milieux poreux. La modélisation est mise en place, étape par étape: organisation spatiale des cristaux d'hydroxyapatite, prise en compte de minéralisations différentes, d'une nouvelle loi de comportement, d'un fluide contenant des ions à chacun des niveaux architecturaux et homogénéisation de structures composites complexes (lamelles, ostéon, os cortical). Sur le plan mathématique, on reprend la méthode des développements asymptotiques dans un cadre piézoélectrique (avec seuil), on établit toutes les relations nécessaires, une propriété de convergence et une estimation de propriétés locales. Le retour au microscopique est fait directement via une technique de localisation ou indirectement lorsque l'effet de seuil se produit. Les méthodes numériques ont été implantées dans deux logiciels. Sur le plan biomécanique, on établit que l'os cortical humain est un milieu orthotrope non piézo électrique pour lequel l'anisotropie est due à l'architecture nanoscopique, que les ostéons sont le siège de deux types d'écoulement, que les écoulements y différent selon l'architecture : on voit comment les cellules savent quelle architecture donner au tissu collagènique

SINUPROS: human cortical bone multiscale model with a fluide-structure interaction

International audienceCells live in a fluid environment located in a biological tissue and the knowledge of the behaviour of this fluid needs the knowledge of the tissue itself. In the present study, the biological tissue is the cortical bone which has a very complex architecture

Human cortical bone: the SiNuPrOs model Part I—description and elastic macroscopic results

International audienceMany models that have been developed for cortical bone oversimplify much of the architectural and physical complexity. With SiNuPrOs model, a more complete approach is investigated: it is multiscale because it contains five structural levels and multi physic because it takes into account simultaneously structure (with various properties: elasticity, piezoelectricity, porous medium), fluid and mineralization process modelization. The multiscale aspect is modeled by using 18 structural parameters in a specific application of the mathematical theory of homogenization and 10 other physical parameters are necessary for the multi physic aspect. The modelization of collagen as a piezoelectric medium has needed the development of a new behaviour law allowing a better simulation of the effect of a medium considered as evolving during a mineralization process. Then the main interest of SiNuPrOs deals with the possibility to study, at each level of the cortical architecture, either the elastic properties or the fluid motion or the piezoelectric effects or both of them. All these possibilities constitute a very large work and all this mass of information (fluid aspects, even at the nanoscopic scale, piezoelectric phenomena and simulations) will be presented in several papers. This first one is only devoted to the presentation of this model with an application to the computation of elastic properties at the macroscopic scale. The computational methods have been packed into software also called SiNuPrOs and allowing a large number of predictive simulations corresponding to various different configurations